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Journal of applied physical science international

DERIVATION OF EVOLUTION WAVE PACKETS BASED ON SCATTERING WAVE FUNCTIONS FOR ONE DIMENSIONAL POTENTIAL

А. Ж. Хачатрян, VOSKANYAN, D. R., KHOECYAN, V. A.

Published Dec 25, 2015• 0 citations
OpenAlex

Abstract

In this work a wave process evolution problem in a field of an arbitrary one-dimensional potential is considered. We investigate the wave packets derived based on linear combinations of scattering wave functions taken with magnitudes of state indexes near to resonance tunneling case. It is shown that the wave processes with different parities of asymptotes describe a wave process where the group and phase velocities have opposite directions. The conducted consideration allows to state that the bound states can be interpreted as wave packets of the scattering wave functions to be derived under a certain way.

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In this work a wave process evolution problem in a field of an arbitrary one-dimensional potential is considered. We investigate the wave packets derived based on linear combinations of scattering wave functions taken with magnitudes of state indexes near to resonance tunneling case. It is shown that the wave processes with different parities of asymptotes describe a wave process where the group and phase velocities have opposite directions. The conducted consideration allows to state that the bound states can be interpreted as wave packets of the scattering wave functions to be derived under a certain way.
In this work a wave process evolution problem in a field of an arbitrary one-dimensional potential is considered. We investigate the wave packets derived based on linear combinations of scattering wave functions taken with magnitudes of state indexes near to resonance tunneling case. It is shown that the wave processes with different parities of asymptotes describe a wave process where the group and phase velocities have opposite directions. The conducted consideration allows to state that the bound states can be interpreted as wave packets of the scattering wave functions to be derived under a certain way.
We consider the evolution of a wave packet that is made up of a group of the wave functions describing the stationary scattering process and tunnels through a one-dimensional potential of arbitrary form. As the main characteristics of the time difference of the tunnelling process, use is made of the propagation speed of the wave-packet maximum. We show that the known Hartman formula for the tunnelling time corresponds to the wave packet with a wavenumber-uniform spectral composition in the case, when the phase and transmission coefficient modulus dispersions are taken into account only in the linear approximation. The amplitude of the main peak of the transmitted wave intensity is proven to be independent of the tunnelling time and is determined by the transmission coefficient of the spectral component at the carrier frequency and the spectral width of the wave packet. In the limit of an infinitely wide potential barrier the amplitude of the wave-packet maximum is shown to tend to zero slower than the tunnelling time tends to its asymptotic value, i.e., indeed we deal with the paradox of an infinitely large propagation speed of a wave disturbance through the barrier.
THE PRESENT NOTE ANALYSES THE DYNAMICAL DESCRIPTION OF THE SCATTERING DUE TO A SHORT RANGE POTENTIAL. IN PARTICULAR, IT DEVELOPS A CLASS OF TRANSFORMS THAT ARE USEFUL FOR THIS DYNAMICAL DESCRIPTION. A TIME-DEPENDENT GREEN FUNCTION FOR THE SCATTERING PROCESS IS DISCUSSED, THAT TAKES US FROM THE INITIAL STATE (PLANE WAVE) TO THE FINAL STATE (PLANE PLUS SCATTERED WAVES). THIS GREEN FUNCTION CAN BE USED TO GIVER THE MOTION OF ANY WAVE PACKET THAT INITIALLY WAS OUTSIDE THE POTENTIAL. THE PRESENT PAPER COMPLEMENTS A PREVIOUS DYNAMICAL DESCRIPTION OF A SCATTERING PROCESS, IN WHICH THE INTERACTION WAS REPRESENTED BY APPROPRIATE BOUNDARY.
It is shown that in the case of the one-particle one-dimensional scattering problem for a given time-independent potential, for each state of the whole quantum ensemble of identically prepared particles, there is an unique pair of (subensemble's) solutions to the Schrödinger equation, which, as we postulate, describe separately transmission and reflection: in the case of nonstationary states, for any instant of time, these functions are orthogonal and their sum describes the state of all particles; evolving with constant norms, one of them approaches at late times the transmitted wave packet and another approaches the reflected packet. Both for transmission and reflection, 1) well before and after the scattering event, the average kinetic energy of particles is the same, 2) the average starting point differs, in the general case, from that for all particles. It is shown that for reflection, in the case of symmetric potential barriers, the domain of the motion of particles is bounded by the midpoint of the barrier region. We define (exact and asymptotic) transmission and reflection times and show that the basic results of our formalism can be, in principle, checked experimentally.
In this paper we consider the time evolution of wave-packet tunneling through a potential barrier. Using a path-summation approach, we can describe the quantum evolution in terms of particle trajectories both in the allowed and forbidden regions. Near the barrier edges, where the potential changes rapidly, the wave aspects dominate. These make the trajectories branch into infinite families of paths, which have to be included in the summation. Combining these features, we treat correctly the complementary aspects of the full quantum process, and this allows us to obtain a computational method that gives accurate numerical results with considerably less expenditure of computer time than a direct integration of the Schr\"odinger equation. We test the method on a simple rectangular potential by comparing the calculations with both conventional wave scattering results and computations from the Schr\"odinger equation. A general potential can be approximated by a staircase, which allows the application of our method. A simple adiabatic extension is shown to work excellently for the case of a periodically modulated barrier.
The normalization problem of a wave function for an arbitrary one-dimensional field is considered. It is shown, that for an infinite motion type the magnitude of the normalization constant does not depend on the form of a scattering potential and it is determined by the boundary conditions imposed on the character of the investigated motion. The general formula for the normalization constant as a function of amplitudes of the waves converging to the potential is derived. Keyword: Normalization problem, wave function, one-dimensional motion, arbitrary potential PACS: 03.65.Nk, 02.30.Em